Interpreting Log-Ratio Effects as Proportion Changes

From a Log-Ratio Coefficient to a Proportion

A fitted coefficient in a multinomial abundance model is not, by itself, a percentage-point change in a cell type’s proportion. It describes a change in a log ratio between that cell type and a reference cell type. To communicate the result on the proportion scale, calculate predicted proportions for the covariate values being compared, using all modeled cell types and the normalization that makes their proportions sum to one.

The model below describes the expected composition conditional on sample-level covariates. Its interpretation concerns relative abundance in the modeled composition. A change in relative abundance does not, on its own, show that the absolute number of cells of a type increased or decreased.

Definition: Let $p_{ik}$ be the modeled proportion of cell type $k$ in sample $i$, among $K$ cell types, and choose cell type $R$ as the reference. A reference-category multinomial log-ratio model specifies, for each $k\ne R$, $$ \log\left(\frac{p_{ik}}{p_{iR}}\right)=\eta_{ik}=x_i^\mathsf{T}\beta_k, $$ where $x_i$ is the sample’s covariate vector and $\beta_k$ is the coefficient vector for cell type $k$ relative to $R$. The proportions satisfy $p_{ik}>0$ and $\sum_{k=1}^{K}p_{ik}=1$.

A common sampling model is that the vector of observed cell-type counts $Y_i=(Y_{i1},\ldots,Y_{iK})$, conditional on the total $N_i$ and covariates, follows a multinomial distribution with probabilities $p_i=(p_{i1},\ldots,p_{iK})$. This formulation assumes independent sample units conditional on the modeled covariates, a suitable conditional mean model for the log ratios, and a sampling-variation model adequate for the data. If counts are more variable than the multinomial model allows, or samples have dependence not represented in the model, uncertainty calculations and possibly the fitted model need to account for that structure. The probability-scale conversion itself applies to the fitted log ratios, but does not remedy a poor model fit.

Recovering the Full Composition

For a specified covariate vector $x$, first calculate each fitted linear predictor $\eta_k(x)$, $k\ne R$. Exponentiating gives the ratio of the predicted proportion for type $k$ to that for the reference type. The reference proportion is then determined by the requirement that all proportions sum to one.

\frac{p_k(x)}{p_R(x)}=\exp\{\eta_k(x)\}, \qquad p_k(x)=p_R(x)\exp\{\eta_k(x)\}.

Substitute the second relationship into the sum-to-one condition. In particular, the sum below is over the nonreference cell types, and its summand has the same index as the cell type being summed.

p_R(x)+\sum_{j\ne R}p_j(x) = p_R(x)\left[1+\sum_{j\ne R}\exp\{\eta_j(x)\}\right] =1.

Solving this equation gives the reference proportion. Multiplying it by each fitted ratio gives every other proportion. These are the reference-category form of the softmax conversion.

p_R(x)=\frac{1}{1+\sum_{j\ne R}\exp\{\eta_j(x)\}}, \qquad p_k(x)=\frac{\exp\{\eta_k(x)\}}{1+\sum_{j\ne R}\exp\{\eta_j(x)\}} \quad (k\ne R).

Key interpretation: If a covariate increases by one unit and its coefficient for type $k$ is $\beta_k$, then the ratio $p_k/p_R$, holding the other modeled covariates fixed and assuming no interaction changes that effect, is multiplied by $\exp(\beta_k)$. This is a multiplier for a ratio of proportions, not a direct multiplier for $p_k$ itself and not a percentage-point change.

Worked Proportion-Scale Contrasts

Worked Example: A Binary Condition and Three Cell Types

Suppose a hypothetical model has cell types A, B, and R, with R as the reference. Let the control-group log ratios be $\log(p_A/p_R)=0$ and $\log(p_B/p_R)=0$. For treatment relative to control, suppose the fitted coefficients are $\log 2$ for A and $0$ for B. There are no other covariates in this example. Thus treatment doubles A’s ratio to R while leaving B’s ratio to R unchanged.

For controls, both ratios are $\exp(0)=1$. The normalizing denominator is $1+1+1=3$, so $p_R=1/3$, $p_A=1/3$, and $p_B=1/3$.

For treatment, the ratios to R are $p_A/p_R=\exp(\log 2)=2$ and $p_B/p_R=\exp(0)=1$. The denominator is $1+2+1=4$, giving $p_R=1/4$, $p_A=2/4=1/2$, and $p_B=1/4$.

The predicted proportion of A is therefore $0.5000-0.3333=0.1667$, an increase of about 16.67 percentage points. Its proportion is 1.5 times the control proportion, although its ratio to R is twice as large. Type B’s ratio to R did not change, yet its predicted proportion fell from $1/3$ to $1/4$. This is a consequence of normalization: the shares must sum to one.

Worked Example: A Continuous Covariate

Consider a hypothetical covariate $x$, measured in years, and three types A, B, and R. At $x=0$, suppose the fitted ratios are $p_A/p_R=0.4$ and $p_B/p_R=0.8$. Let the coefficient per year be $\log(1.5)$ for A and $\log(0.75)$ for B. The model has linear predictors $\eta_A(x)=\log(0.4)+x\log(1.5)$ and $\eta_B(x)=\log(0.8)+x\log(0.75)$.

At $x=0$, the denominator is $1+0.4+0.8=2.2$. Therefore $p_R=1/2.2=0.4545$, $p_A=0.4/2.2=0.1818$, and $p_B=0.8/2.2=0.3636$.

At $x=2$, the fitted ratios are $0.4(1.5)^2=0.9000$ for A and $0.8(0.75)^2=0.4500$ for B. The denominator is $1+0.9+0.45=2.35$. Thus $p_R=1/2.35=0.4255$, $p_A=0.9/2.35=0.3830$, and $p_B=0.45/2.35=0.1915$, to four significant figures.

For a two-year increase, A’s ratio to R is multiplied by $(1.5)^2=2.25$, while its predicted proportion increases by $0.3830-0.1818=0.2012$, or about 20.12 percentage points. These are different quantities. The proportion contrast is specific to the comparison from $x=0$ to $x=2$; it is not a constant increase per year implied by the log-ratio coefficient.

Reporting the Contrast and Its Limits

A clear report names the comparison, the reference category, the covariate values held fixed, and the scale of each result. For example: “At the specified covariate values, the model predicts that A comprises 50.0% of the composition under treatment and 33.3% under control, a difference of 16.7 percentage points. The treatment-to-control multiplier for A’s ratio to reference type R is 2.” This distinguishes the model coefficient’s scale from the predicted-proportion scale.

For a treatment indicator in a model without a treatment interaction, the coefficient for type $k$ is the treatment-versus-control difference in $\log(p_k/p_R)$ at fixed values of the other covariates. Exponentiating gives the corresponding ratio-of-ratios. The predicted proportions still depend on the fitted linear predictors for all nonreference types. If the model contains interactions, the treatment contrast can depend on the values of the interacting covariates; compute the linear predictors at the specific values rather than reading a single coefficient as a universal treatment effect.

Proportions describe composition, not the total number of cells in a tissue or sample. For example, if a hypothetical control sample has 1,000 cells with A at one third, its expected A count is about 333. If a hypothetical treated sample has 400 cells with A at one half, its expected A count is 200. A’s proportion is higher in the second sample, while its count is lower. Conversely, other total counts could produce a higher A count at the same predicted proportion. Counts of captured cells also need not equal absolute tissue abundance, since collection and capture can affect what is observed.

Worked Example: Reporting an Adjusted Comparison

Suppose a hypothetical fitted model uses R as reference and includes a binary treatment indicator $T$ and a covariate $z$, with no interaction. For type A, let $\eta_A=-1+(\log 2)T+0.02z$; for type B, let $\eta_B=-0.5+0.01z$. Compare treatment with control at $z=10$, holding $z$ fixed.

For control, $\eta_A=-1+0+0.2=-0.8$, so $p_A/p_R=\exp(-0.8)=0.4493$. For treatment, $\eta_A=-1+\log 2+0.2\approx-0.1069$, so $p_A/p_R=\exp(-1+\log 2+0.2)\approx0.8987$. The ratio of these ratios is $0.8987/0.4493=2.000$, as also given by $\exp(\log 2)=2$.

For both groups, $\eta_B=-0.5+0.1=-0.4$, so $p_B/p_R=\exp(-0.4)=0.6703$. The control denominator is $1+0.4493+0.6703=2.1196$, giving $p_A=0.4493/2.1196=0.2120$. The treatment denominator is $1+0.8987+0.6703=2.5690$, giving $p_A=0.8987/2.5690=0.3498$. The model-predicted A proportion rises by $0.3498-0.2120=0.1378$, about 13.78 percentage points, at $z=10$. A complete report gives both this adjusted proportion contrast and the ratio multiplier, while identifying the reference type and the covariate value.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting converted proportions, check that the fitted model and the intended contrast are appropriate. Confirm that the reference category and covariate coding are understood, that the requested values are within a sensible range of the observed data, and that every fitted proportion is positive and the proportions sum to one, allowing for rounding. Examine residual and overdispersion diagnostics or other model checks appropriate to the fitted approach. Large systematic discrepancies between observed and predicted compositions, sparse categories, influential samples, or unexplained sample dependence can make the model-based predictions unreliable.

Predicted proportions and log-ratio coefficients answer complementary questions. The coefficient is compact and directly describes a relative ratio change; the predicted proportions show the resulting composition at a specified covariate profile. Reporting only one can obscure the other. A component-wise comparison may instead focus on a single type, but should not be described as a complete composition contrast unless it accounts for the other components and the model’s conditioning. In all cases, the result remains an association in modeled relative abundance, not evidence of a causal effect or of an absolute cell-number change.

Mistake: “A coefficient of $\log 2$ means the cell type’s proportion doubles.” Correction: it means the cell type’s proportion relative to the reference proportion is multiplied by two for the specified covariate change. Calculate normalized predicted proportions to describe the share change.
Mistake: “The reference ratio for B is unchanged, so B’s proportion is unchanged.” Correction: other ratios can change, altering the normalizing denominator and therefore B’s share.
Mistake: “A higher predicted proportion proves more cells of that type.” Correction: absolute counts depend on the total number of cells, and observed capture totals may not measure total tissue abundance.
Mistake: “A coefficient is the same at every covariate value.” Correction: this is only a direct interpretation when the model form supports it. With interactions, calculate the contrast at stated covariate values.

Key takeaway: Exponentiate a log-ratio contrast to describe the multiplier in a cell-type-to-reference ratio. To report a proportion-scale effect, predict and normalize the full composition at each comparison value, then state the difference in proportions and the covariate context. Do not infer absolute cell gains from relative proportions alone.

Check Your Understanding

Answer each question, then check the brief solutions.

With three types A, B, and R, the fitted ratios at one covariate value are $p_A/p_R=1$ and $p_B/p_R=3$. What are the three proportions?
A one-unit covariate increase has coefficient $\log(0.5)$ for A relative to R. What does this say about the ratio, and what does it not establish about A’s proportion?
If a modeled composition changes from A = 0.30 to A = 0.45, state the difference in percentage points. Does that alone establish an increase in A’s absolute count?
Why can B’s proportion change even when its fitted ratio to R does not change?

Solutions. (1) The denominator is $1+1+3=5$, so $p_R=1/5=0.20$, $p_A=1/5=0.20$, and $p_B=3/5=0.60$. (2) The ratio $p_A/p_R$ is multiplied by $\exp\{\log(0.5)\}=0.5$; this does not specify A’s proportion change without the other fitted ratios and the starting composition. (3) The difference is $0.45-0.30=0.15$, or 15 percentage points. The total cell count is needed to compare counts, and observed counts may not equal absolute tissue abundance. (4) The normalization denominator depends on the ratios for all nonreference types, so changing another ratio can alter the normalized proportions of both B and R even if $p_B/p_R$ stays fixed.